Does a non-uniformly-dense pyramid floating in water displace more water/“float lower” than a uniformly dense one?

Consider two square, equilateral pyramids of equal volume $$V_{object}$$ and equal mass.

1. One pyramid is uniformly dense
2. The other pyramid has most of its mass concentrated at the apex.

Both pyramids have an (equal) overall density $$\rho_{object}$$ less than that of water ($$\rho_{fluid}$$), so they float. Will the pyramids float with a different amount submerged, and if so, why?

On one hand, I know that objects float when the gravitational force on the object is equal and opposite to the buoyant force of the water on the object. This implies that the mass of the object and the displaced water are equal, or

$$\rho_{fluid} V_{displaced} = \rho_{object} V_{object}.$$ Since my rearranging the location of the mass in the object without changing its density, we know that $$\rho_{fluid}$$, $$\rho_{object}$$, and $$V_{object}$$ are all held constant, so the volume displaced must remain the same.

On the other hand perhaps the fact that the denser part of the pyramid will likely face down when submerged means that the PART of the pyramid submerged in the water is denser, so we need to increase $$\rho_{object}$$ in this equation. That would imply that the $$V_{displaced}$$ would have to increase to compensate.

Is one of these arguments correct, or are they both wrong?